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June 8, 2024 00:20
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InverseTrigonometry by solidity-trigonometry. Find it at https://www.cookbook.dev/contracts/solidity-trigonometry-InverseTrigonometry
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/* | |
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Find any smart contract, and build your project faster: https://www.cookbook.dev | |
Twitter: https://twitter.com/cookbook_dev | |
Discord: https://discord.gg/WzsfPcfHrk | |
Find this contract on Cookbook: https://www.cookbook.dev/contracts/solidity-trigonometry-InverseTrigonometry/?utm=code | |
*/ | |
// SPDX-License-Identifier: MIT | |
pragma solidity ^0.8.13; | |
import {PRBMathSD59x18 as P} from "./PRBMathSD59x18.sol"; | |
/** | |
* @title Arcsine calculator. | |
* @author Md Abid Sikder | |
* | |
* @notice Calculates arcsine. Fuzz testing shows that relative error is always | |
* smaller than 0.01%. Uses the polynomial approximation functions found in | |
* https://dsp.stackexchange.com/a/25771, but chooses between them at x=0.4788 | |
* due to differences in the relative errors as can be seen here | |
* https://www.desmos.com/calculator/wrfwjhythe | |
* | |
* @dev See the desmos link for what functions f and g in the code refer to. | |
*/ | |
library InverseTrigonometry { | |
using P for int256; | |
function g(int256 _x) internal pure returns (int256) { | |
int256 ONE = 1000000000000000000; | |
int256 TWO = 2000000000000000000; | |
// 1.5707288 | |
int256 a0 = 1570728800000000000; | |
// −0.2121144 | |
int256 a1 = -212114400000000000; | |
// 0.0742610 | |
int256 a2 = 74261000000000000; | |
// −0.0187293 | |
int256 a3 = -18729300000000000; | |
int256 HALF_PI = P.pi().div(TWO); | |
int256 root = P.sqrt(ONE - _x); | |
return HALF_PI - root.mul(a0 + _x.mul(a1 + _x.mul(a2 + _x.mul(a3)))); | |
} | |
function f(int256 _x) internal pure returns (int256) { | |
int256 ONE = 1000000000000000000; | |
int256 xSq = _x.mul(_x); | |
// 1/6 | |
// https://www.wolframalpha.com/input?i=1%2F6 | |
// 0.1666666666666666666666666 | |
int256 frac1Div6 = 166666666666666666; | |
// 3/40 | |
// https://www.wolframalpha.com/input?i=3%2F40 | |
// 0.075 | |
int256 frac3Div40 = 75000000000000000; | |
// 15/336 | |
// https://www.wolframalpha.com/input?i=15%2F336 | |
// 0.044642857142857142857142857142 | |
int256 frac15Div336= 44642857142857142; | |
return _x.mul(ONE + xSq.mul(frac1Div6 + xSq.mul(frac3Div40 + xSq.mul(frac15Div336)))); | |
} | |
/** | |
* @notice Arcsine function | |
* | |
* @param _x A integer with 18 fixed decimal points, where the whole part is bounded inside of [-1,1] | |
* | |
* @return The arcsine, with 18 fixed decimal points | |
*/ | |
function arcsin(int256 _x) internal pure returns (int256) { | |
int256 DOMAIN_MAX = 1000000000000000000; | |
int256 DOMAIN_MIN = -DOMAIN_MAX; | |
require(_x >= DOMAIN_MIN && _x <= DOMAIN_MAX); | |
// arcsin is an odd function, so arcsin(-x) = -arcsin(x), so we can remove | |
// the negative here for easier math | |
bool isNegative = _x < 0; | |
_x = isNegative ? -_x : _x; | |
// 0.4788 | |
int256 CHOICE_LINE = 478800000000000000; | |
int256 result = _x < CHOICE_LINE ? f(_x) : g(_x); | |
return isNegative ? -result : result; | |
} | |
} |
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/* | |
██████ ██████ ██████ ██ ██ ██████ ██████ ██████ ██ ██ ██████ ███████ ██ ██ | |
██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ | |
██ ██ ██ ██ ██ █████ ██████ ██ ██ ██ ██ █████ ██ ██ █████ ██ ██ | |
██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ | |
██████ ██████ ██████ ██ ██ ██████ ██████ ██████ ██ ██ ██ ██████ ███████ ████ | |
Find any smart contract, and build your project faster: https://www.cookbook.dev | |
Twitter: https://twitter.com/cookbook_dev | |
Discord: https://discord.gg/WzsfPcfHrk | |
Find this contract on Cookbook: https://www.cookbook.dev/contracts/solidity-trigonometry-InverseTrigonometry/?utm=code | |
*/ | |
// SPDX-License-Identifier: Unlicense | |
pragma solidity >=0.8.4; | |
/// @notice Emitted when the result overflows uint256. | |
error PRBMath__MulDivFixedPointOverflow(uint256 prod1); | |
/// @notice Emitted when the result overflows uint256. | |
error PRBMath__MulDivOverflow(uint256 prod1, uint256 denominator); | |
/// @notice Emitted when one of the inputs is type(int256).min. | |
error PRBMath__MulDivSignedInputTooSmall(); | |
/// @notice Emitted when the intermediary absolute result overflows int256. | |
error PRBMath__MulDivSignedOverflow(uint256 rAbs); | |
/// @notice Emitted when the input is MIN_SD59x18. | |
error PRBMathSD59x18__AbsInputTooSmall(); | |
/// @notice Emitted when ceiling a number overflows SD59x18. | |
error PRBMathSD59x18__CeilOverflow(int256 x); | |
/// @notice Emitted when one of the inputs is MIN_SD59x18. | |
error PRBMathSD59x18__DivInputTooSmall(); | |
/// @notice Emitted when one of the intermediary unsigned results overflows SD59x18. | |
error PRBMathSD59x18__DivOverflow(uint256 rAbs); | |
/// @notice Emitted when the input is greater than 133.084258667509499441. | |
error PRBMathSD59x18__ExpInputTooBig(int256 x); | |
/// @notice Emitted when the input is greater than 192. | |
error PRBMathSD59x18__Exp2InputTooBig(int256 x); | |
/// @notice Emitted when flooring a number underflows SD59x18. | |
error PRBMathSD59x18__FloorUnderflow(int256 x); | |
/// @notice Emitted when converting a basic integer to the fixed-point format overflows SD59x18. | |
error PRBMathSD59x18__FromIntOverflow(int256 x); | |
/// @notice Emitted when converting a basic integer to the fixed-point format underflows SD59x18. | |
error PRBMathSD59x18__FromIntUnderflow(int256 x); | |
/// @notice Emitted when the product of the inputs is negative. | |
error PRBMathSD59x18__GmNegativeProduct(int256 x, int256 y); | |
/// @notice Emitted when multiplying the inputs overflows SD59x18. | |
error PRBMathSD59x18__GmOverflow(int256 x, int256 y); | |
/// @notice Emitted when the input is less than or equal to zero. | |
error PRBMathSD59x18__LogInputTooSmall(int256 x); | |
/// @notice Emitted when one of the inputs is MIN_SD59x18. | |
error PRBMathSD59x18__MulInputTooSmall(); | |
/// @notice Emitted when the intermediary absolute result overflows SD59x18. | |
error PRBMathSD59x18__MulOverflow(uint256 rAbs); | |
/// @notice Emitted when the intermediary absolute result overflows SD59x18. | |
error PRBMathSD59x18__PowuOverflow(uint256 rAbs); | |
/// @notice Emitted when the input is negative. | |
error PRBMathSD59x18__SqrtNegativeInput(int256 x); | |
/// @notice Emitted when the calculating the square root overflows SD59x18. | |
error PRBMathSD59x18__SqrtOverflow(int256 x); | |
/// @notice Emitted when addition overflows UD60x18. | |
error PRBMathUD60x18__AddOverflow(uint256 x, uint256 y); | |
/// @notice Emitted when ceiling a number overflows UD60x18. | |
error PRBMathUD60x18__CeilOverflow(uint256 x); | |
/// @notice Emitted when the input is greater than 133.084258667509499441. | |
error PRBMathUD60x18__ExpInputTooBig(uint256 x); | |
/// @notice Emitted when the input is greater than 192. | |
error PRBMathUD60x18__Exp2InputTooBig(uint256 x); | |
/// @notice Emitted when converting a basic integer to the fixed-point format format overflows UD60x18. | |
error PRBMathUD60x18__FromUintOverflow(uint256 x); | |
/// @notice Emitted when multiplying the inputs overflows UD60x18. | |
error PRBMathUD60x18__GmOverflow(uint256 x, uint256 y); | |
/// @notice Emitted when the input is less than 1. | |
error PRBMathUD60x18__LogInputTooSmall(uint256 x); | |
/// @notice Emitted when the calculating the square root overflows UD60x18. | |
error PRBMathUD60x18__SqrtOverflow(uint256 x); | |
/// @notice Emitted when subtraction underflows UD60x18. | |
error PRBMathUD60x18__SubUnderflow(uint256 x, uint256 y); | |
/// @dev Common mathematical functions used in both PRBMathSD59x18 and PRBMathUD60x18. Note that this shared library | |
/// does not always assume the signed 59.18-decimal fixed-point or the unsigned 60.18-decimal fixed-point | |
/// representation. When it does not, it is explicitly mentioned in the NatSpec documentation. | |
library PRBMath { | |
/// STRUCTS /// | |
struct SD59x18 { | |
int256 value; | |
} | |
struct UD60x18 { | |
uint256 value; | |
} | |
/// STORAGE /// | |
/// @dev How many trailing decimals can be represented. | |
uint256 internal constant SCALE = 1e18; | |
/// @dev Largest power of two divisor of SCALE. | |
uint256 internal constant SCALE_LPOTD = 262144; | |
/// @dev SCALE inverted mod 2^256. | |
uint256 internal constant SCALE_INVERSE = | |
78156646155174841979727994598816262306175212592076161876661_508869554232690281; | |
/// FUNCTIONS /// | |
/// @notice Calculates the binary exponent of x using the binary fraction method. | |
/// @dev Has to use 192.64-bit fixed-point numbers. | |
/// See https://ethereum.stackexchange.com/a/96594/24693. | |
/// @param x The exponent as an unsigned 192.64-bit fixed-point number. | |
/// @return result The result as an unsigned 60.18-decimal fixed-point number. | |
function exp2(uint256 x) internal pure returns (uint256 result) { | |
unchecked { | |
// Start from 0.5 in the 192.64-bit fixed-point format. | |
result = 0x800000000000000000000000000000000000000000000000; | |
// Multiply the result by root(2, 2^-i) when the bit at position i is 1. None of the intermediary results overflows | |
// because the initial result is 2^191 and all magic factors are less than 2^65. | |
if (x & 0x8000000000000000 > 0) { | |
result = (result * 0x16A09E667F3BCC909) >> 64; | |
} | |
if (x & 0x4000000000000000 > 0) { | |
result = (result * 0x1306FE0A31B7152DF) >> 64; | |
} | |
if (x & 0x2000000000000000 > 0) { | |
result = (result * 0x1172B83C7D517ADCE) >> 64; | |
} | |
if (x & 0x1000000000000000 > 0) { | |
result = (result * 0x10B5586CF9890F62A) >> 64; | |
} | |
if (x & 0x800000000000000 > 0) { | |
result = (result * 0x1059B0D31585743AE) >> 64; | |
} | |
if (x & 0x400000000000000 > 0) { | |
result = (result * 0x102C9A3E778060EE7) >> 64; | |
} | |
if (x & 0x200000000000000 > 0) { | |
result = (result * 0x10163DA9FB33356D8) >> 64; | |
} | |
if (x & 0x100000000000000 > 0) { | |
result = (result * 0x100B1AFA5ABCBED61) >> 64; | |
} | |
if (x & 0x80000000000000 > 0) { | |
result = (result * 0x10058C86DA1C09EA2) >> 64; | |
} | |
if (x & 0x40000000000000 > 0) { | |
result = (result * 0x1002C605E2E8CEC50) >> 64; | |
} | |
if (x & 0x20000000000000 > 0) { | |
result = (result * 0x100162F3904051FA1) >> 64; | |
} | |
if (x & 0x10000000000000 > 0) { | |
result = (result * 0x1000B175EFFDC76BA) >> 64; | |
} | |
if (x & 0x8000000000000 > 0) { | |
result = (result * 0x100058BA01FB9F96D) >> 64; | |
} | |
if (x & 0x4000000000000 > 0) { | |
result = (result * 0x10002C5CC37DA9492) >> 64; | |
} | |
if (x & 0x2000000000000 > 0) { | |
result = (result * 0x1000162E525EE0547) >> 64; | |
} | |
if (x & 0x1000000000000 > 0) { | |
result = (result * 0x10000B17255775C04) >> 64; | |
} | |
if (x & 0x800000000000 > 0) { | |
result = (result * 0x1000058B91B5BC9AE) >> 64; | |
} | |
if (x & 0x400000000000 > 0) { | |
result = (result * 0x100002C5C89D5EC6D) >> 64; | |
} | |
if (x & 0x200000000000 > 0) { | |
result = (result * 0x10000162E43F4F831) >> 64; | |
} | |
if (x & 0x100000000000 > 0) { | |
result = (result * 0x100000B1721BCFC9A) >> 64; | |
} | |
if (x & 0x80000000000 > 0) { | |
result = (result * 0x10000058B90CF1E6E) >> 64; | |
} | |
if (x & 0x40000000000 > 0) { | |
result = (result * 0x1000002C5C863B73F) >> 64; | |
} | |
if (x & 0x20000000000 > 0) { | |
result = (result * 0x100000162E430E5A2) >> 64; | |
} | |
if (x & 0x10000000000 > 0) { | |
result = (result * 0x1000000B172183551) >> 64; | |
} | |
if (x & 0x8000000000 > 0) { | |
result = (result * 0x100000058B90C0B49) >> 64; | |
} | |
if (x & 0x4000000000 > 0) { | |
result = (result * 0x10000002C5C8601CC) >> 64; | |
} | |
if (x & 0x2000000000 > 0) { | |
result = (result * 0x1000000162E42FFF0) >> 64; | |
} | |
if (x & 0x1000000000 > 0) { | |
result = (result * 0x10000000B17217FBB) >> 64; | |
} | |
if (x & 0x800000000 > 0) { | |
result = (result * 0x1000000058B90BFCE) >> 64; | |
} | |
if (x & 0x400000000 > 0) { | |
result = (result * 0x100000002C5C85FE3) >> 64; | |
} | |
if (x & 0x200000000 > 0) { | |
result = (result * 0x10000000162E42FF1) >> 64; | |
} | |
if (x & 0x100000000 > 0) { | |
result = (result * 0x100000000B17217F8) >> 64; | |
} | |
if (x & 0x80000000 > 0) { | |
result = (result * 0x10000000058B90BFC) >> 64; | |
} | |
if (x & 0x40000000 > 0) { | |
result = (result * 0x1000000002C5C85FE) >> 64; | |
} | |
if (x & 0x20000000 > 0) { | |
result = (result * 0x100000000162E42FF) >> 64; | |
} | |
if (x & 0x10000000 > 0) { | |
result = (result * 0x1000000000B17217F) >> 64; | |
} | |
if (x & 0x8000000 > 0) { | |
result = (result * 0x100000000058B90C0) >> 64; | |
} | |
if (x & 0x4000000 > 0) { | |
result = (result * 0x10000000002C5C860) >> 64; | |
} | |
if (x & 0x2000000 > 0) { | |
result = (result * 0x1000000000162E430) >> 64; | |
} | |
if (x & 0x1000000 > 0) { | |
result = (result * 0x10000000000B17218) >> 64; | |
} | |
if (x & 0x800000 > 0) { | |
result = (result * 0x1000000000058B90C) >> 64; | |
} | |
if (x & 0x400000 > 0) { | |
result = (result * 0x100000000002C5C86) >> 64; | |
} | |
if (x & 0x200000 > 0) { | |
result = (result * 0x10000000000162E43) >> 64; | |
} | |
if (x & 0x100000 > 0) { | |
result = (result * 0x100000000000B1721) >> 64; | |
} | |
if (x & 0x80000 > 0) { | |
result = (result * 0x10000000000058B91) >> 64; | |
} | |
if (x & 0x40000 > 0) { | |
result = (result * 0x1000000000002C5C8) >> 64; | |
} | |
if (x & 0x20000 > 0) { | |
result = (result * 0x100000000000162E4) >> 64; | |
} | |
if (x & 0x10000 > 0) { | |
result = (result * 0x1000000000000B172) >> 64; | |
} | |
if (x & 0x8000 > 0) { | |
result = (result * 0x100000000000058B9) >> 64; | |
} | |
if (x & 0x4000 > 0) { | |
result = (result * 0x10000000000002C5D) >> 64; | |
} | |
if (x & 0x2000 > 0) { | |
result = (result * 0x1000000000000162E) >> 64; | |
} | |
if (x & 0x1000 > 0) { | |
result = (result * 0x10000000000000B17) >> 64; | |
} | |
if (x & 0x800 > 0) { | |
result = (result * 0x1000000000000058C) >> 64; | |
} | |
if (x & 0x400 > 0) { | |
result = (result * 0x100000000000002C6) >> 64; | |
} | |
if (x & 0x200 > 0) { | |
result = (result * 0x10000000000000163) >> 64; | |
} | |
if (x & 0x100 > 0) { | |
result = (result * 0x100000000000000B1) >> 64; | |
} | |
if (x & 0x80 > 0) { | |
result = (result * 0x10000000000000059) >> 64; | |
} | |
if (x & 0x40 > 0) { | |
result = (result * 0x1000000000000002C) >> 64; | |
} | |
if (x & 0x20 > 0) { | |
result = (result * 0x10000000000000016) >> 64; | |
} | |
if (x & 0x10 > 0) { | |
result = (result * 0x1000000000000000B) >> 64; | |
} | |
if (x & 0x8 > 0) { | |
result = (result * 0x10000000000000006) >> 64; | |
} | |
if (x & 0x4 > 0) { | |
result = (result * 0x10000000000000003) >> 64; | |
} | |
if (x & 0x2 > 0) { | |
result = (result * 0x10000000000000001) >> 64; | |
} | |
if (x & 0x1 > 0) { | |
result = (result * 0x10000000000000001) >> 64; | |
} | |
// We're doing two things at the same time: | |
// | |
// 1. Multiply the result by 2^n + 1, where "2^n" is the integer part and the one is added to account for | |
// the fact that we initially set the result to 0.5. This is accomplished by subtracting from 191 | |
// rather than 192. | |
// 2. Convert the result to the unsigned 60.18-decimal fixed-point format. | |
// | |
// This works because 2^(191-ip) = 2^ip / 2^191, where "ip" is the integer part "2^n". | |
result *= SCALE; | |
result >>= (191 - (x >> 64)); | |
} | |
} | |
/// @notice Finds the zero-based index of the first one in the binary representation of x. | |
/// @dev See the note on msb in the "Find First Set" Wikipedia article https://en.wikipedia.org/wiki/Find_first_set | |
/// @param x The uint256 number for which to find the index of the most significant bit. | |
/// @return msb The index of the most significant bit as an uint256. | |
function mostSignificantBit(uint256 x) internal pure returns (uint256 msb) { | |
if (x >= 2**128) { | |
x >>= 128; | |
msb += 128; | |
} | |
if (x >= 2**64) { | |
x >>= 64; | |
msb += 64; | |
} | |
if (x >= 2**32) { | |
x >>= 32; | |
msb += 32; | |
} | |
if (x >= 2**16) { | |
x >>= 16; | |
msb += 16; | |
} | |
if (x >= 2**8) { | |
x >>= 8; | |
msb += 8; | |
} | |
if (x >= 2**4) { | |
x >>= 4; | |
msb += 4; | |
} | |
if (x >= 2**2) { | |
x >>= 2; | |
msb += 2; | |
} | |
if (x >= 2**1) { | |
// No need to shift x any more. | |
msb += 1; | |
} | |
} | |
/// @notice Calculates floor(x*y÷denominator) with full precision. | |
/// | |
/// @dev Credit to Remco Bloemen under MIT license https://xn--2-umb.com/21/muldiv. | |
/// | |
/// Requirements: | |
/// - The denominator cannot be zero. | |
/// - The result must fit within uint256. | |
/// | |
/// Caveats: | |
/// - This function does not work with fixed-point numbers. | |
/// | |
/// @param x The multiplicand as an uint256. | |
/// @param y The multiplier as an uint256. | |
/// @param denominator The divisor as an uint256. | |
/// @return result The result as an uint256. | |
function mulDiv( | |
uint256 x, | |
uint256 y, | |
uint256 denominator | |
) internal pure returns (uint256 result) { | |
// 512-bit multiply [prod1 prod0] = x * y. Compute the product mod 2^256 and mod 2^256 - 1, then use | |
// use the Chinese Remainder Theorem to reconstruct the 512 bit result. The result is stored in two 256 | |
// variables such that product = prod1 * 2^256 + prod0. | |
uint256 prod0; // Least significant 256 bits of the product | |
uint256 prod1; // Most significant 256 bits of the product | |
assembly { | |
let mm := mulmod(x, y, not(0)) | |
prod0 := mul(x, y) | |
prod1 := sub(sub(mm, prod0), lt(mm, prod0)) | |
} | |
// Handle non-overflow cases, 256 by 256 division. | |
if (prod1 == 0) { | |
unchecked { | |
result = prod0 / denominator; | |
} | |
return result; | |
} | |
// Make sure the result is less than 2^256. Also prevents denominator == 0. | |
if (prod1 >= denominator) { | |
revert PRBMath__MulDivOverflow(prod1, denominator); | |
} | |
/////////////////////////////////////////////// | |
// 512 by 256 division. | |
/////////////////////////////////////////////// | |
// Make division exact by subtracting the remainder from [prod1 prod0]. | |
uint256 remainder; | |
assembly { | |
// Compute remainder using mulmod. | |
remainder := mulmod(x, y, denominator) | |
// Subtract 256 bit number from 512 bit number. | |
prod1 := sub(prod1, gt(remainder, prod0)) | |
prod0 := sub(prod0, remainder) | |
} | |
// Factor powers of two out of denominator and compute largest power of two divisor of denominator. Always >= 1. | |
// See https://cs.stackexchange.com/q/138556/92363. | |
unchecked { | |
// Does not overflow because the denominator cannot be zero at this stage in the function. | |
uint256 lpotdod = denominator & (~denominator + 1); | |
assembly { | |
// Divide denominator by lpotdod. | |
denominator := div(denominator, lpotdod) | |
// Divide [prod1 prod0] by lpotdod. | |
prod0 := div(prod0, lpotdod) | |
// Flip lpotdod such that it is 2^256 / lpotdod. If lpotdod is zero, then it becomes one. | |
lpotdod := add(div(sub(0, lpotdod), lpotdod), 1) | |
} | |
// Shift in bits from prod1 into prod0. | |
prod0 |= prod1 * lpotdod; | |
// Invert denominator mod 2^256. Now that denominator is an odd number, it has an inverse modulo 2^256 such | |
// that denominator * inv = 1 mod 2^256. Compute the inverse by starting with a seed that is correct for | |
// four bits. That is, denominator * inv = 1 mod 2^4. | |
uint256 inverse = (3 * denominator) ^ 2; | |
// Use the Newton-Raphson iteration to improve the precision. Thanks to Hensel's lifting lemma, this also works | |
// in modular arithmetic, doubling the correct bits in each step. | |
inverse *= 2 - denominator * inverse; // inverse mod 2^8 | |
inverse *= 2 - denominator * inverse; // inverse mod 2^16 | |
inverse *= 2 - denominator * inverse; // inverse mod 2^32 | |
inverse *= 2 - denominator * inverse; // inverse mod 2^64 | |
inverse *= 2 - denominator * inverse; // inverse mod 2^128 | |
inverse *= 2 - denominator * inverse; // inverse mod 2^256 | |
// Because the division is now exact we can divide by multiplying with the modular inverse of denominator. | |
// This will give us the correct result modulo 2^256. Since the preconditions guarantee that the outcome is | |
// less than 2^256, this is the final result. We don't need to compute the high bits of the result and prod1 | |
// is no longer required. | |
result = prod0 * inverse; | |
return result; | |
} | |
} | |
/// @notice Calculates floor(x*y÷1e18) with full precision. | |
/// | |
/// @dev Variant of "mulDiv" with constant folding, i.e. in which the denominator is always 1e18. Before returning the | |
/// final result, we add 1 if (x * y) % SCALE >= HALF_SCALE. Without this, 6.6e-19 would be truncated to 0 instead of | |
/// being rounded to 1e-18. See "Listing 6" and text above it at https://accu.org/index.php/journals/1717. | |
/// | |
/// Requirements: | |
/// - The result must fit within uint256. | |
/// | |
/// Caveats: | |
/// - The body is purposely left uncommented; see the NatSpec comments in "PRBMath.mulDiv" to understand how this works. | |
/// - It is assumed that the result can never be type(uint256).max when x and y solve the following two equations: | |
/// 1. x * y = type(uint256).max * SCALE | |
/// 2. (x * y) % SCALE >= SCALE / 2 | |
/// | |
/// @param x The multiplicand as an unsigned 60.18-decimal fixed-point number. | |
/// @param y The multiplier as an unsigned 60.18-decimal fixed-point number. | |
/// @return result The result as an unsigned 60.18-decimal fixed-point number. | |
function mulDivFixedPoint(uint256 x, uint256 y) internal pure returns (uint256 result) { | |
uint256 prod0; | |
uint256 prod1; | |
assembly { | |
let mm := mulmod(x, y, not(0)) | |
prod0 := mul(x, y) | |
prod1 := sub(sub(mm, prod0), lt(mm, prod0)) | |
} | |
if (prod1 >= SCALE) { | |
revert PRBMath__MulDivFixedPointOverflow(prod1); | |
} | |
uint256 remainder; | |
uint256 roundUpUnit; | |
assembly { | |
remainder := mulmod(x, y, SCALE) | |
roundUpUnit := gt(remainder, 499999999999999999) | |
} | |
if (prod1 == 0) { | |
unchecked { | |
result = (prod0 / SCALE) + roundUpUnit; | |
return result; | |
} | |
} | |
assembly { | |
result := add( | |
mul( | |
or( | |
div(sub(prod0, remainder), SCALE_LPOTD), | |
mul(sub(prod1, gt(remainder, prod0)), add(div(sub(0, SCALE_LPOTD), SCALE_LPOTD), 1)) | |
), | |
SCALE_INVERSE | |
), | |
roundUpUnit | |
) | |
} | |
} | |
/// @notice Calculates floor(x*y÷denominator) with full precision. | |
/// | |
/// @dev An extension of "mulDiv" for signed numbers. Works by computing the signs and the absolute values separately. | |
/// | |
/// Requirements: | |
/// - None of the inputs can be type(int256).min. | |
/// - The result must fit within int256. | |
/// | |
/// @param x The multiplicand as an int256. | |
/// @param y The multiplier as an int256. | |
/// @param denominator The divisor as an int256. | |
/// @return result The result as an int256. | |
function mulDivSigned( | |
int256 x, | |
int256 y, | |
int256 denominator | |
) internal pure returns (int256 result) { | |
if (x == type(int256).min || y == type(int256).min || denominator == type(int256).min) { | |
revert PRBMath__MulDivSignedInputTooSmall(); | |
} | |
// Get hold of the absolute values of x, y and the denominator. | |
uint256 ax; | |
uint256 ay; | |
uint256 ad; | |
unchecked { | |
ax = x < 0 ? uint256(-x) : uint256(x); | |
ay = y < 0 ? uint256(-y) : uint256(y); | |
ad = denominator < 0 ? uint256(-denominator) : uint256(denominator); | |
} | |
// Compute the absolute value of (x*y)÷denominator. The result must fit within int256. | |
uint256 rAbs = mulDiv(ax, ay, ad); | |
if (rAbs > uint256(type(int256).max)) { | |
revert PRBMath__MulDivSignedOverflow(rAbs); | |
} | |
// Get the signs of x, y and the denominator. | |
uint256 sx; | |
uint256 sy; | |
uint256 sd; | |
assembly { | |
sx := sgt(x, sub(0, 1)) | |
sy := sgt(y, sub(0, 1)) | |
sd := sgt(denominator, sub(0, 1)) | |
} | |
// XOR over sx, sy and sd. This is checking whether there are one or three negative signs in the inputs. | |
// If yes, the result should be negative. | |
result = sx ^ sy ^ sd == 0 ? -int256(rAbs) : int256(rAbs); | |
} | |
/// @notice Calculates the square root of x, rounding down. | |
/// @dev Uses the Babylonian method https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method. | |
/// | |
/// Caveats: | |
/// - This function does not work with fixed-point numbers. | |
/// | |
/// @param x The uint256 number for which to calculate the square root. | |
/// @return result The result as an uint256. | |
function sqrt(uint256 x) internal pure returns (uint256 result) { | |
if (x == 0) { | |
return 0; | |
} | |
// Set the initial guess to the least power of two that is greater than or equal to sqrt(x). | |
uint256 xAux = uint256(x); | |
result = 1; | |
if (xAux >= 0x100000000000000000000000000000000) { | |
xAux >>= 128; | |
result <<= 64; | |
} | |
if (xAux >= 0x10000000000000000) { | |
xAux >>= 64; | |
result <<= 32; | |
} | |
if (xAux >= 0x100000000) { | |
xAux >>= 32; | |
result <<= 16; | |
} | |
if (xAux >= 0x10000) { | |
xAux >>= 16; | |
result <<= 8; | |
} | |
if (xAux >= 0x100) { | |
xAux >>= 8; | |
result <<= 4; | |
} | |
if (xAux >= 0x10) { | |
xAux >>= 4; | |
result <<= 2; | |
} | |
if (xAux >= 0x4) { | |
result <<= 1; | |
} | |
// The operations can never overflow because the result is max 2^127 when it enters this block. | |
unchecked { | |
result = (result + x / result) >> 1; | |
result = (result + x / result) >> 1; | |
result = (result + x / result) >> 1; | |
result = (result + x / result) >> 1; | |
result = (result + x / result) >> 1; | |
result = (result + x / result) >> 1; | |
result = (result + x / result) >> 1; // Seven iterations should be enough | |
uint256 roundedDownResult = x / result; | |
return result >= roundedDownResult ? roundedDownResult : result; | |
} | |
} | |
} |
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/* | |
██████ ██████ ██████ ██ ██ ██████ ██████ ██████ ██ ██ ██████ ███████ ██ ██ | |
██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ | |
██ ██ ██ ██ ██ █████ ██████ ██ ██ ██ ██ █████ ██ ██ █████ ██ ██ | |
██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ ██ | |
██████ ██████ ██████ ██ ██ ██████ ██████ ██████ ██ ██ ██ ██████ ███████ ████ | |
Find any smart contract, and build your project faster: https://www.cookbook.dev | |
Twitter: https://twitter.com/cookbook_dev | |
Discord: https://discord.gg/WzsfPcfHrk | |
Find this contract on Cookbook: https://www.cookbook.dev/contracts/solidity-trigonometry-InverseTrigonometry/?utm=code | |
*/ | |
// SPDX-License-Identifier: Unlicense | |
pragma solidity >=0.8.4; | |
import "./PRBMath.sol"; | |
/// @title PRBMathSD59x18 | |
/// @author Paul Razvan Berg | |
/// @notice Smart contract library for advanced fixed-point math that works with int256 numbers considered to have 18 | |
/// trailing decimals. We call this number representation signed 59.18-decimal fixed-point, since the numbers can have | |
/// a sign and there can be up to 59 digits in the integer part and up to 18 decimals in the fractional part. The numbers | |
/// are bound by the minimum and the maximum values permitted by the Solidity type int256. | |
library PRBMathSD59x18 { | |
/// @dev log2(e) as a signed 59.18-decimal fixed-point number. | |
int256 internal constant LOG2_E = 1_442695040888963407; | |
/// @dev Half the SCALE number. | |
int256 internal constant HALF_SCALE = 5e17; | |
/// @dev The maximum value a signed 59.18-decimal fixed-point number can have. | |
int256 internal constant MAX_SD59x18 = | |
57896044618658097711785492504343953926634992332820282019728_792003956564819967; | |
/// @dev The maximum whole value a signed 59.18-decimal fixed-point number can have. | |
int256 internal constant MAX_WHOLE_SD59x18 = | |
57896044618658097711785492504343953926634992332820282019728_000000000000000000; | |
/// @dev The minimum value a signed 59.18-decimal fixed-point number can have. | |
int256 internal constant MIN_SD59x18 = | |
-57896044618658097711785492504343953926634992332820282019728_792003956564819968; | |
/// @dev The minimum whole value a signed 59.18-decimal fixed-point number can have. | |
int256 internal constant MIN_WHOLE_SD59x18 = | |
-57896044618658097711785492504343953926634992332820282019728_000000000000000000; | |
/// @dev How many trailing decimals can be represented. | |
int256 internal constant SCALE = 1e18; | |
/// INTERNAL FUNCTIONS /// | |
/// @notice Calculate the absolute value of x. | |
/// | |
/// @dev Requirements: | |
/// - x must be greater than MIN_SD59x18. | |
/// | |
/// @param x The number to calculate the absolute value for. | |
/// @param result The absolute value of x. | |
function abs(int256 x) internal pure returns (int256 result) { | |
unchecked { | |
if (x == MIN_SD59x18) { | |
revert PRBMathSD59x18__AbsInputTooSmall(); | |
} | |
result = x < 0 ? -x : x; | |
} | |
} | |
/// @notice Calculates the arithmetic average of x and y, rounding down. | |
/// @param x The first operand as a signed 59.18-decimal fixed-point number. | |
/// @param y The second operand as a signed 59.18-decimal fixed-point number. | |
/// @return result The arithmetic average as a signed 59.18-decimal fixed-point number. | |
function avg(int256 x, int256 y) internal pure returns (int256 result) { | |
// The operations can never overflow. | |
unchecked { | |
int256 sum = (x >> 1) + (y >> 1); | |
if (sum < 0) { | |
// If at least one of x and y is odd, we add 1 to the result. This is because shifting negative numbers to the | |
// right rounds down to infinity. | |
assembly { | |
result := add(sum, and(or(x, y), 1)) | |
} | |
} else { | |
// If both x and y are odd, we add 1 to the result. This is because if both numbers are odd, the 0.5 | |
// remainder gets truncated twice. | |
result = sum + (x & y & 1); | |
} | |
} | |
} | |
/// @notice Yields the least greatest signed 59.18 decimal fixed-point number greater than or equal to x. | |
/// | |
/// @dev Optimized for fractional value inputs, because for every whole value there are (1e18 - 1) fractional counterparts. | |
/// See https://en.wikipedia.org/wiki/Floor_and_ceiling_functions. | |
/// | |
/// Requirements: | |
/// - x must be less than or equal to MAX_WHOLE_SD59x18. | |
/// | |
/// @param x The signed 59.18-decimal fixed-point number to ceil. | |
/// @param result The least integer greater than or equal to x, as a signed 58.18-decimal fixed-point number. | |
function ceil(int256 x) internal pure returns (int256 result) { | |
if (x > MAX_WHOLE_SD59x18) { | |
revert PRBMathSD59x18__CeilOverflow(x); | |
} | |
unchecked { | |
int256 remainder = x % SCALE; | |
if (remainder == 0) { | |
result = x; | |
} else { | |
// Solidity uses C fmod style, which returns a modulus with the same sign as x. | |
result = x - remainder; | |
if (x > 0) { | |
result += SCALE; | |
} | |
} | |
} | |
} | |
/// @notice Divides two signed 59.18-decimal fixed-point numbers, returning a new signed 59.18-decimal fixed-point number. | |
/// | |
/// @dev Variant of "mulDiv" that works with signed numbers. Works by computing the signs and the absolute values separately. | |
/// | |
/// Requirements: | |
/// - All from "PRBMath.mulDiv". | |
/// - None of the inputs can be MIN_SD59x18. | |
/// - The denominator cannot be zero. | |
/// - The result must fit within int256. | |
/// | |
/// Caveats: | |
/// - All from "PRBMath.mulDiv". | |
/// | |
/// @param x The numerator as a signed 59.18-decimal fixed-point number. | |
/// @param y The denominator as a signed 59.18-decimal fixed-point number. | |
/// @param result The quotient as a signed 59.18-decimal fixed-point number. | |
function div(int256 x, int256 y) internal pure returns (int256 result) { | |
if (x == MIN_SD59x18 || y == MIN_SD59x18) { | |
revert PRBMathSD59x18__DivInputTooSmall(); | |
} | |
// Get hold of the absolute values of x and y. | |
uint256 ax; | |
uint256 ay; | |
unchecked { | |
ax = x < 0 ? uint256(-x) : uint256(x); | |
ay = y < 0 ? uint256(-y) : uint256(y); | |
} | |
// Compute the absolute value of (x*SCALE)÷y. The result must fit within int256. | |
uint256 rAbs = PRBMath.mulDiv(ax, uint256(SCALE), ay); | |
if (rAbs > uint256(MAX_SD59x18)) { | |
revert PRBMathSD59x18__DivOverflow(rAbs); | |
} | |
// Get the signs of x and y. | |
uint256 sx; | |
uint256 sy; | |
assembly { | |
sx := sgt(x, sub(0, 1)) | |
sy := sgt(y, sub(0, 1)) | |
} | |
// XOR over sx and sy. This is basically checking whether the inputs have the same sign. If yes, the result | |
// should be positive. Otherwise, it should be negative. | |
result = sx ^ sy == 1 ? -int256(rAbs) : int256(rAbs); | |
} | |
/// @notice Returns Euler's number as a signed 59.18-decimal fixed-point number. | |
/// @dev See https://en.wikipedia.org/wiki/E_(mathematical_constant). | |
function e() internal pure returns (int256 result) { | |
result = 2_718281828459045235; | |
} | |
/// @notice Calculates the natural exponent of x. | |
/// | |
/// @dev Based on the insight that e^x = 2^(x * log2(e)). | |
/// | |
/// Requirements: | |
/// - All from "log2". | |
/// - x must be less than 133.084258667509499441. | |
/// | |
/// Caveats: | |
/// - All from "exp2". | |
/// - For any x less than -41.446531673892822322, the result is zero. | |
/// | |
/// @param x The exponent as a signed 59.18-decimal fixed-point number. | |
/// @return result The result as a signed 59.18-decimal fixed-point number. | |
function exp(int256 x) internal pure returns (int256 result) { | |
// Without this check, the value passed to "exp2" would be less than -59.794705707972522261. | |
if (x < -41_446531673892822322) { | |
return 0; | |
} | |
// Without this check, the value passed to "exp2" would be greater than 192. | |
if (x >= 133_084258667509499441) { | |
revert PRBMathSD59x18__ExpInputTooBig(x); | |
} | |
// Do the fixed-point multiplication inline to save gas. | |
unchecked { | |
int256 doubleScaleProduct = x * LOG2_E; | |
result = exp2((doubleScaleProduct + HALF_SCALE) / SCALE); | |
} | |
} | |
/// @notice Calculates the binary exponent of x using the binary fraction method. | |
/// | |
/// @dev See https://ethereum.stackexchange.com/q/79903/24693. | |
/// | |
/// Requirements: | |
/// - x must be 192 or less. | |
/// - The result must fit within MAX_SD59x18. | |
/// | |
/// Caveats: | |
/// - For any x less than -59.794705707972522261, the result is zero. | |
/// | |
/// @param x The exponent as a signed 59.18-decimal fixed-point number. | |
/// @return result The result as a signed 59.18-decimal fixed-point number. | |
function exp2(int256 x) internal pure returns (int256 result) { | |
// This works because 2^(-x) = 1/2^x. | |
if (x < 0) { | |
// 2^59.794705707972522262 is the maximum number whose inverse does not truncate down to zero. | |
if (x < -59_794705707972522261) { | |
return 0; | |
} | |
// Do the fixed-point inversion inline to save gas. The numerator is SCALE * SCALE. | |
unchecked { | |
result = 1e36 / exp2(-x); | |
} | |
} else { | |
// 2^192 doesn't fit within the 192.64-bit format used internally in this function. | |
if (x >= 192e18) { | |
revert PRBMathSD59x18__Exp2InputTooBig(x); | |
} | |
unchecked { | |
// Convert x to the 192.64-bit fixed-point format. | |
uint256 x192x64 = (uint256(x) << 64) / uint256(SCALE); | |
// Safe to convert the result to int256 directly because the maximum input allowed is 192. | |
result = int256(PRBMath.exp2(x192x64)); | |
} | |
} | |
} | |
/// @notice Yields the greatest signed 59.18 decimal fixed-point number less than or equal to x. | |
/// | |
/// @dev Optimized for fractional value inputs, because for every whole value there are (1e18 - 1) fractional counterparts. | |
/// See https://en.wikipedia.org/wiki/Floor_and_ceiling_functions. | |
/// | |
/// Requirements: | |
/// - x must be greater than or equal to MIN_WHOLE_SD59x18. | |
/// | |
/// @param x The signed 59.18-decimal fixed-point number to floor. | |
/// @param result The greatest integer less than or equal to x, as a signed 58.18-decimal fixed-point number. | |
function floor(int256 x) internal pure returns (int256 result) { | |
if (x < MIN_WHOLE_SD59x18) { | |
revert PRBMathSD59x18__FloorUnderflow(x); | |
} | |
unchecked { | |
int256 remainder = x % SCALE; | |
if (remainder == 0) { | |
result = x; | |
} else { | |
// Solidity uses C fmod style, which returns a modulus with the same sign as x. | |
result = x - remainder; | |
if (x < 0) { | |
result -= SCALE; | |
} | |
} | |
} | |
} | |
/// @notice Yields the excess beyond the floor of x for positive numbers and the part of the number to the right | |
/// of the radix point for negative numbers. | |
/// @dev Based on the odd function definition. https://en.wikipedia.org/wiki/Fractional_part | |
/// @param x The signed 59.18-decimal fixed-point number to get the fractional part of. | |
/// @param result The fractional part of x as a signed 59.18-decimal fixed-point number. | |
function frac(int256 x) internal pure returns (int256 result) { | |
unchecked { | |
result = x % SCALE; | |
} | |
} | |
/// @notice Converts a number from basic integer form to signed 59.18-decimal fixed-point representation. | |
/// | |
/// @dev Requirements: | |
/// - x must be greater than or equal to MIN_SD59x18 divided by SCALE. | |
/// - x must be less than or equal to MAX_SD59x18 divided by SCALE. | |
/// | |
/// @param x The basic integer to convert. | |
/// @param result The same number in signed 59.18-decimal fixed-point representation. | |
function fromInt(int256 x) internal pure returns (int256 result) { | |
unchecked { | |
if (x < MIN_SD59x18 / SCALE) { | |
revert PRBMathSD59x18__FromIntUnderflow(x); | |
} | |
if (x > MAX_SD59x18 / SCALE) { | |
revert PRBMathSD59x18__FromIntOverflow(x); | |
} | |
result = x * SCALE; | |
} | |
} | |
/// @notice Calculates geometric mean of x and y, i.e. sqrt(x * y), rounding down. | |
/// | |
/// @dev Requirements: | |
/// - x * y must fit within MAX_SD59x18, lest it overflows. | |
/// - x * y cannot be negative. | |
/// | |
/// @param x The first operand as a signed 59.18-decimal fixed-point number. | |
/// @param y The second operand as a signed 59.18-decimal fixed-point number. | |
/// @return result The result as a signed 59.18-decimal fixed-point number. | |
function gm(int256 x, int256 y) internal pure returns (int256 result) { | |
if (x == 0) { | |
return 0; | |
} | |
unchecked { | |
// Checking for overflow this way is faster than letting Solidity do it. | |
int256 xy = x * y; | |
if (xy / x != y) { | |
revert PRBMathSD59x18__GmOverflow(x, y); | |
} | |
// The product cannot be negative. | |
if (xy < 0) { | |
revert PRBMathSD59x18__GmNegativeProduct(x, y); | |
} | |
// We don't need to multiply by the SCALE here because the x*y product had already picked up a factor of SCALE | |
// during multiplication. See the comments within the "sqrt" function. | |
result = int256(PRBMath.sqrt(uint256(xy))); | |
} | |
} | |
/// @notice Calculates 1 / x, rounding toward zero. | |
/// | |
/// @dev Requirements: | |
/// - x cannot be zero. | |
/// | |
/// @param x The signed 59.18-decimal fixed-point number for which to calculate the inverse. | |
/// @return result The inverse as a signed 59.18-decimal fixed-point number. | |
function inv(int256 x) internal pure returns (int256 result) { | |
unchecked { | |
// 1e36 is SCALE * SCALE. | |
result = 1e36 / x; | |
} | |
} | |
/// @notice Calculates the natural logarithm of x. | |
/// | |
/// @dev Based on the insight that ln(x) = log2(x) / log2(e). | |
/// | |
/// Requirements: | |
/// - All from "log2". | |
/// | |
/// Caveats: | |
/// - All from "log2". | |
/// - This doesn't return exactly 1 for 2718281828459045235, for that we would need more fine-grained precision. | |
/// | |
/// @param x The signed 59.18-decimal fixed-point number for which to calculate the natural logarithm. | |
/// @return result The natural logarithm as a signed 59.18-decimal fixed-point number. | |
function ln(int256 x) internal pure returns (int256 result) { | |
// Do the fixed-point multiplication inline to save gas. This is overflow-safe because the maximum value that log2(x) | |
// can return is 195205294292027477728. | |
unchecked { | |
result = (log2(x) * SCALE) / LOG2_E; | |
} | |
} | |
/// @notice Calculates the common logarithm of x. | |
/// | |
/// @dev First checks if x is an exact power of ten and it stops if yes. If it's not, calculates the common | |
/// logarithm based on the insight that log10(x) = log2(x) / log2(10). | |
/// | |
/// Requirements: | |
/// - All from "log2". | |
/// | |
/// Caveats: | |
/// - All from "log2". | |
/// | |
/// @param x The signed 59.18-decimal fixed-point number for which to calculate the common logarithm. | |
/// @return result The common logarithm as a signed 59.18-decimal fixed-point number. | |
function log10(int256 x) internal pure returns (int256 result) { | |
if (x <= 0) { | |
revert PRBMathSD59x18__LogInputTooSmall(x); | |
} | |
// Note that the "mul" in this block is the assembly mul operation, not the "mul" function defined in this contract. | |
// prettier-ignore | |
assembly { | |
switch x | |
case 1 { result := mul(SCALE, sub(0, 18)) } | |
case 10 { result := mul(SCALE, sub(1, 18)) } | |
case 100 { result := mul(SCALE, sub(2, 18)) } | |
case 1000 { result := mul(SCALE, sub(3, 18)) } | |
case 10000 { result := mul(SCALE, sub(4, 18)) } | |
case 100000 { result := mul(SCALE, sub(5, 18)) } | |
case 1000000 { result := mul(SCALE, sub(6, 18)) } | |
case 10000000 { result := mul(SCALE, sub(7, 18)) } | |
case 100000000 { result := mul(SCALE, sub(8, 18)) } | |
case 1000000000 { result := mul(SCALE, sub(9, 18)) } | |
case 10000000000 { result := mul(SCALE, sub(10, 18)) } | |
case 100000000000 { result := mul(SCALE, sub(11, 18)) } | |
case 1000000000000 { result := mul(SCALE, sub(12, 18)) } | |
case 10000000000000 { result := mul(SCALE, sub(13, 18)) } | |
case 100000000000000 { result := mul(SCALE, sub(14, 18)) } | |
case 1000000000000000 { result := mul(SCALE, sub(15, 18)) } | |
case 10000000000000000 { result := mul(SCALE, sub(16, 18)) } | |
case 100000000000000000 { result := mul(SCALE, sub(17, 18)) } | |
case 1000000000000000000 { result := 0 } | |
case 10000000000000000000 { result := SCALE } | |
case 100000000000000000000 { result := mul(SCALE, 2) } | |
case 1000000000000000000000 { result := mul(SCALE, 3) } | |
case 10000000000000000000000 { result := mul(SCALE, 4) } | |
case 100000000000000000000000 { result := mul(SCALE, 5) } | |
case 1000000000000000000000000 { result := mul(SCALE, 6) } | |
case 10000000000000000000000000 { result := mul(SCALE, 7) } | |
case 100000000000000000000000000 { result := mul(SCALE, 8) } | |
case 1000000000000000000000000000 { result := mul(SCALE, 9) } | |
case 10000000000000000000000000000 { result := mul(SCALE, 10) } | |
case 100000000000000000000000000000 { result := mul(SCALE, 11) } | |
case 1000000000000000000000000000000 { result := mul(SCALE, 12) } | |
case 10000000000000000000000000000000 { result := mul(SCALE, 13) } | |
case 100000000000000000000000000000000 { result := mul(SCALE, 14) } | |
case 1000000000000000000000000000000000 { result := mul(SCALE, 15) } | |
case 10000000000000000000000000000000000 { result := mul(SCALE, 16) } | |
case 100000000000000000000000000000000000 { result := mul(SCALE, 17) } | |
case 1000000000000000000000000000000000000 { result := mul(SCALE, 18) } | |
case 10000000000000000000000000000000000000 { result := mul(SCALE, 19) } | |
case 100000000000000000000000000000000000000 { result := mul(SCALE, 20) } | |
case 1000000000000000000000000000000000000000 { result := mul(SCALE, 21) } | |
case 10000000000000000000000000000000000000000 { result := mul(SCALE, 22) } | |
case 100000000000000000000000000000000000000000 { result := mul(SCALE, 23) } | |
case 1000000000000000000000000000000000000000000 { result := mul(SCALE, 24) } | |
case 10000000000000000000000000000000000000000000 { result := mul(SCALE, 25) } | |
case 100000000000000000000000000000000000000000000 { result := mul(SCALE, 26) } | |
case 1000000000000000000000000000000000000000000000 { result := mul(SCALE, 27) } | |
case 10000000000000000000000000000000000000000000000 { result := mul(SCALE, 28) } | |
case 100000000000000000000000000000000000000000000000 { result := mul(SCALE, 29) } | |
case 1000000000000000000000000000000000000000000000000 { result := mul(SCALE, 30) } | |
case 10000000000000000000000000000000000000000000000000 { result := mul(SCALE, 31) } | |
case 100000000000000000000000000000000000000000000000000 { result := mul(SCALE, 32) } | |
case 1000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 33) } | |
case 10000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 34) } | |
case 100000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 35) } | |
case 1000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 36) } | |
case 10000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 37) } | |
case 100000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 38) } | |
case 1000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 39) } | |
case 10000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 40) } | |
case 100000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 41) } | |
case 1000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 42) } | |
case 10000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 43) } | |
case 100000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 44) } | |
case 1000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 45) } | |
case 10000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 46) } | |
case 100000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 47) } | |
case 1000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 48) } | |
case 10000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 49) } | |
case 100000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 50) } | |
case 1000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 51) } | |
case 10000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 52) } | |
case 100000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 53) } | |
case 1000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 54) } | |
case 10000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 55) } | |
case 100000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 56) } | |
case 1000000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 57) } | |
case 10000000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 58) } | |
default { | |
result := MAX_SD59x18 | |
} | |
} | |
if (result == MAX_SD59x18) { | |
// Do the fixed-point division inline to save gas. The denominator is log2(10). | |
unchecked { | |
result = (log2(x) * SCALE) / 3_321928094887362347; | |
} | |
} | |
} | |
/// @notice Calculates the binary logarithm of x. | |
/// | |
/// @dev Based on the iterative approximation algorithm. | |
/// https://en.wikipedia.org/wiki/Binary_logarithm#Iterative_approximation | |
/// | |
/// Requirements: | |
/// - x must be greater than zero. | |
/// | |
/// Caveats: | |
/// - The results are not perfectly accurate to the last decimal, due to the lossy precision of the iterative approximation. | |
/// | |
/// @param x The signed 59.18-decimal fixed-point number for which to calculate the binary logarithm. | |
/// @return result The binary logarithm as a signed 59.18-decimal fixed-point number. | |
function log2(int256 x) internal pure returns (int256 result) { | |
if (x <= 0) { | |
revert PRBMathSD59x18__LogInputTooSmall(x); | |
} | |
unchecked { | |
// This works because log2(x) = -log2(1/x). | |
int256 sign; | |
if (x >= SCALE) { | |
sign = 1; | |
} else { | |
sign = -1; | |
// Do the fixed-point inversion inline to save gas. The numerator is SCALE * SCALE. | |
assembly { | |
x := div(1000000000000000000000000000000000000, x) | |
} | |
} | |
// Calculate the integer part of the logarithm and add it to the result and finally calculate y = x * 2^(-n). | |
uint256 n = PRBMath.mostSignificantBit(uint256(x / SCALE)); | |
// The integer part of the logarithm as a signed 59.18-decimal fixed-point number. The operation can't overflow | |
// because n is maximum 255, SCALE is 1e18 and sign is either 1 or -1. | |
result = int256(n) * SCALE; | |
// This is y = x * 2^(-n). | |
int256 y = x >> n; | |
// If y = 1, the fractional part is zero. | |
if (y == SCALE) { | |
return result * sign; | |
} | |
// Calculate the fractional part via the iterative approximation. | |
// The "delta >>= 1" part is equivalent to "delta /= 2", but shifting bits is faster. | |
for (int256 delta = int256(HALF_SCALE); delta > 0; delta >>= 1) { | |
y = (y * y) / SCALE; | |
// Is y^2 > 2 and so in the range [2,4)? | |
if (y >= 2 * SCALE) { | |
// Add the 2^(-m) factor to the logarithm. | |
result += delta; | |
// Corresponds to z/2 on Wikipedia. | |
y >>= 1; | |
} | |
} | |
result *= sign; | |
} | |
} | |
/// @notice Multiplies two signed 59.18-decimal fixed-point numbers together, returning a new signed 59.18-decimal | |
/// fixed-point number. | |
/// | |
/// @dev Variant of "mulDiv" that works with signed numbers and employs constant folding, i.e. the denominator is | |
/// always 1e18. | |
/// | |
/// Requirements: | |
/// - All from "PRBMath.mulDivFixedPoint". | |
/// - None of the inputs can be MIN_SD59x18 | |
/// - The result must fit within MAX_SD59x18. | |
/// | |
/// Caveats: | |
/// - The body is purposely left uncommented; see the NatSpec comments in "PRBMath.mulDiv" to understand how this works. | |
/// | |
/// @param x The multiplicand as a signed 59.18-decimal fixed-point number. | |
/// @param y The multiplier as a signed 59.18-decimal fixed-point number. | |
/// @return result The product as a signed 59.18-decimal fixed-point number. | |
function mul(int256 x, int256 y) internal pure returns (int256 result) { | |
if (x == MIN_SD59x18 || y == MIN_SD59x18) { | |
revert PRBMathSD59x18__MulInputTooSmall(); | |
} | |
unchecked { | |
uint256 ax; | |
uint256 ay; | |
ax = x < 0 ? uint256(-x) : uint256(x); | |
ay = y < 0 ? uint256(-y) : uint256(y); | |
uint256 rAbs = PRBMath.mulDivFixedPoint(ax, ay); | |
if (rAbs > uint256(MAX_SD59x18)) { | |
revert PRBMathSD59x18__MulOverflow(rAbs); | |
} | |
uint256 sx; | |
uint256 sy; | |
assembly { | |
sx := sgt(x, sub(0, 1)) | |
sy := sgt(y, sub(0, 1)) | |
} | |
result = sx ^ sy == 1 ? -int256(rAbs) : int256(rAbs); | |
} | |
} | |
/// @notice Returns PI as a signed 59.18-decimal fixed-point number. | |
function pi() internal pure returns (int256 result) { | |
result = 3_141592653589793238; | |
} | |
/// @notice Raises x to the power of y. | |
/// | |
/// @dev Based on the insight that x^y = 2^(log2(x) * y). | |
/// | |
/// Requirements: | |
/// - All from "exp2", "log2" and "mul". | |
/// - z cannot be zero. | |
/// | |
/// Caveats: | |
/// - All from "exp2", "log2" and "mul". | |
/// - Assumes 0^0 is 1. | |
/// | |
/// @param x Number to raise to given power y, as a signed 59.18-decimal fixed-point number. | |
/// @param y Exponent to raise x to, as a signed 59.18-decimal fixed-point number. | |
/// @return result x raised to power y, as a signed 59.18-decimal fixed-point number. | |
function pow(int256 x, int256 y) internal pure returns (int256 result) { | |
if (x == 0) { | |
result = y == 0 ? SCALE : int256(0); | |
} else { | |
result = exp2(mul(log2(x), y)); | |
} | |
} | |
/// @notice Raises x (signed 59.18-decimal fixed-point number) to the power of y (basic unsigned integer) using the | |
/// famous algorithm "exponentiation by squaring". | |
/// | |
/// @dev See https://en.wikipedia.org/wiki/Exponentiation_by_squaring | |
/// | |
/// Requirements: | |
/// - All from "abs" and "PRBMath.mulDivFixedPoint". | |
/// - The result must fit within MAX_SD59x18. | |
/// | |
/// Caveats: | |
/// - All from "PRBMath.mulDivFixedPoint". | |
/// - Assumes 0^0 is 1. | |
/// | |
/// @param x The base as a signed 59.18-decimal fixed-point number. | |
/// @param y The exponent as an uint256. | |
/// @return result The result as a signed 59.18-decimal fixed-point number. | |
function powu(int256 x, uint256 y) internal pure returns (int256 result) { | |
uint256 xAbs = uint256(abs(x)); | |
// Calculate the first iteration of the loop in advance. | |
uint256 rAbs = y & 1 > 0 ? xAbs : uint256(SCALE); | |
// Equivalent to "for(y /= 2; y > 0; y /= 2)" but faster. | |
uint256 yAux = y; | |
for (yAux >>= 1; yAux > 0; yAux >>= 1) { | |
xAbs = PRBMath.mulDivFixedPoint(xAbs, xAbs); | |
// Equivalent to "y % 2 == 1" but faster. | |
if (yAux & 1 > 0) { | |
rAbs = PRBMath.mulDivFixedPoint(rAbs, xAbs); | |
} | |
} | |
// The result must fit within the 59.18-decimal fixed-point representation. | |
if (rAbs > uint256(MAX_SD59x18)) { | |
revert PRBMathSD59x18__PowuOverflow(rAbs); | |
} | |
// Is the base negative and the exponent an odd number? | |
bool isNegative = x < 0 && y & 1 == 1; | |
result = isNegative ? -int256(rAbs) : int256(rAbs); | |
} | |
/// @notice Returns 1 as a signed 59.18-decimal fixed-point number. | |
function scale() internal pure returns (int256 result) { | |
result = SCALE; | |
} | |
/// @notice Calculates the square root of x, rounding down. | |
/// @dev Uses the Babylonian method https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method. | |
/// | |
/// Requirements: | |
/// - x cannot be negative. | |
/// - x must be less than MAX_SD59x18 / SCALE. | |
/// | |
/// @param x The signed 59.18-decimal fixed-point number for which to calculate the square root. | |
/// @return result The result as a signed 59.18-decimal fixed-point . | |
function sqrt(int256 x) internal pure returns (int256 result) { | |
unchecked { | |
if (x < 0) { | |
revert PRBMathSD59x18__SqrtNegativeInput(x); | |
} | |
if (x > MAX_SD59x18 / SCALE) { | |
revert PRBMathSD59x18__SqrtOverflow(x); | |
} | |
// Multiply x by the SCALE to account for the factor of SCALE that is picked up when multiplying two signed | |
// 59.18-decimal fixed-point numbers together (in this case, those two numbers are both the square root). | |
result = int256(PRBMath.sqrt(uint256(x * SCALE))); | |
} | |
} | |
/// @notice Converts a signed 59.18-decimal fixed-point number to basic integer form, rounding down in the process. | |
/// @param x The signed 59.18-decimal fixed-point number to convert. | |
/// @return result The same number in basic integer form. | |
function toInt(int256 x) internal pure returns (int256 result) { | |
unchecked { | |
result = x / SCALE; | |
} | |
} | |
} |
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